A TALE OF TWO ARC LENGTHS: METRIC NOTIONS FOR CURVES IN SURFACES IN EQUIAFFINE SPACE

Michael Thomas Schmidt, Jeanne Clelland, edward estrada, Molly May, Jonah Miller, Sean Peneyra

Abstract:

In Euclidean geometry, all metric notions (arc length for curves,
the first fundamental form for surfaces, etc.) are derived from the Euclidean
inner product on tangent vectors, and this inner product is preserved by the full
symmetry group of Euclidean space (translations, rotations, and reflections).
In equiaffine geometry, there is no invariant notion of inner product on tangent
vectors that is preserved by the full equiaffine symmetry group. Nevertheless,
it is possible to define an invariant notion of arc length for nondegenerate
curves and an invariant first fundamental form for nondegenerate surfaces in
equiaffine space. This leads to two possible notions of arc length for a curve
contained in a surface, and these two arc length functions do not necessarily
agree. In this paper we will derive necessary and sufficient conditions under
which the two arc length functions do agree, and illustrate with examples.